Chua&#39;s circuit and it&#39;s use in hyperchaotic circuit

ABSTRACT

The present invention provides an improved Chua&#39;s circuit providing current mode operation, access to all state variables, minimum use of grounded passive elements, and freedom from passive component matching comprising a dual output current conveyer based inductor having one grounded terminal, a capacitor connected across the second terminal of said inductor, a resistor having one terminal connected to the second terminal of said inductor, the second terminal of said resistor connected to one terminal of a second capacitor the other end of which is grounded, and a pair of dual output current conveyers connected together to form a 2-terminal negative resistance having one terminal connected to ground and the second terminal connected to the second terminal of said resistance.

FIELD OF THE INVENTION

The invention relates to an improved Chua's circuit more particularly,the invention relates to a Chua's circuit using a Dual Output CurrentConveyer (DO-CC II). The invention also implements Chua's circuit usingMultiple Output Current Conveyor (MO-CCII) and uses the said designedcircuit to generate a hyperchaotic circuit.

BACKGROUND OF THE INVENTION

FIG. 1 shows a generalized Chua's circuit that has a parallelcombination of a capacitor, C₂, a Chua's diode, connected to an L-C tankcircuit that consists of a second capacitor C₁ connected in parallelwith a serially connected inductor, L₁. In addition, a first voltage, V₁is defined across capacitor C₁ and second voltage, V₂ defined acrosscapacitor C₂ with the positive orientation of voltages V₁ and V₂ bothbeing at the positive end of the corresponding capacitor. Finally, thereis a linear resistor R₁ connected between the positive terminals of eachcapacitor.

In a typical configuration, the non-linear curve is piece-wise linearwith symmetrical slope discontinuities around current axis. It satisfiesthe following equationI_(R)=G_(a)V_(R)+(½)(G_(a)−G_(b)){|v_(R)+B_(p)|−|v_(R)−B_(p)|} whereG_(a) and G_(b) are the slopes of respective linear portions ofpiece-wise linear current/voltage curve characterizing the nonlinearresistor and B_(p) is the absolute value of the two voltage points atwhich discontinuities in the current/voltage curve lie as shown in FIG.2. The circuit has a circuit driving subsystem, a L-C tank circuit, anda response subsystem with a parallel combination of capacitor andnon-linear resistor interconnected through resistor.

By choosing values of R, L, C₁ and C₂, the circuit can operate indifferent operating regions for example double scroll region. HereinChua's circuit can be made that will oscillate chaotically orquasi-periodically. Given a specified physical configuration and aspecified initial state specified by V₁, V₂ and I_(L) the voltage acrossthe capacitors C₁ and C₂ and the current through the inductor L, theevolution of Chua's circuit is deterministic, but chaotic. That is anyChua's circuit with the same physical parameters and initial conditionswill follow the same course of states over time and this course willrepeat itself over a very long period. However, to an observer it looksmainly like a noise. Also, systems trajectory is sensitive to initialconditions. In addition, the power spectral density function is spreadover a wide range of frequencies, with the peak frequency of thefundamental being governed by L-C tank circuit.

Owing to its simple circuitry, ability to demonstrate most well knownroutes to chaos, Chua's Circuit and Chua's Oscillator is an active topicof research in the study of non-linear dynamical circuit and systems.Recently, there has been an increasing interest in designinginductor-less Chua's circuit and Chua's oscillator. Moreover, owing toadvantages, the attainability of all the three state variables of CC/COis also attracting the designers. Simultaneously literature is alsowitnessing the shift of analog integrated circuit designing from voltagemode processing to current-mode processing (CMP). A number of chaoticcircuits have been implemented using current-mode active buildingblocks. Dual-Output Current Conveyor (DO-CC II) is also emerging as aversatile block to implement current-mode circuits.

To improve the performance of the circuit different researchers havedone extensive research. A paper by Kennedy M. P. [Kennedy M. P.;‘Robust op-amp realization of Chua's circuit’, Frequenz, 1992, 46, pp.66-80.] suggests a Chua's circuit using off shelf components. Further,in another research paper by Torres, L. A. B. and Aguirre, L. A.[Torres, L. A. B. and Aguirre, L. A.; Inductorless Chua's circuit,Electron. Lett., 2000, 36, (23), pp. 1915-1916.] report a Chua's circuitusing operational amplifier to generate Chua's oscillation at very lowfrequency that is claimed to be used for bio-medical operations.

In a design proposed by Morgul, O. [Morgul, O.; ‘Inductorlessrealization of Chua's oscillator’, Electron. Lett., 1995, 31, pp.1403-1404.] synthetic inductors (using op-amps) were used along with theoperational amplifier thereby making design suitable for monolithicimplementation.

Senani R. and Gupta S. S. [Senani R. and Gupta S. S.; ‘Implementation ofChua's chaotic circuit using current feedback op-amps’, Electron. Lett.,1998, 34, (9), pp. 829-830.] proposed Chua's circuit using CurrentFeedback Operational Amplifier (CFOA) thus making available the thirdstate variable through the inductor (i_(L)).

In another architecture designed by Elwakil A. S. and Kennedy M. P.[Elwakil A. S. and Kennedy M. P.; ‘Improved implementation of Chua'schaotic oscillator using current-feedback op-amp’, IEEE Trans. CAS-I,2000, 47, (1), pp. 76- 79.] CFOA was efficiently used in Chua's circuitto provide a higher bandwidth of chaotic signal with buffered output ofone state variable.

All the above architectures individually provide some or the otheradvantage of Chua's circuit but so far there is no circuit thatsimultaneously provides:

-   -   1. Current mode operation;    -   2. Use of minimum grounded passive elements;    -   3. Availability of all the state variables;    -   4. Availability of two state variables in form of current which        further can be used further to generate other complex chaotic        circuits;    -   5. A circuit idea free from passive component matching;    -   6. Use of lesser active components as compared to prior art 2.    -   7. Generation of reduced hardware higher order chaotic circuit        (also called hyper-chaotic circuit) using one of the available        current mentioned in 4 above.

Thus it is observed that there is a need to develop a circuit that canprovide above all simultaneously.

Background of Application in Generating Hyper Chaos

Chua's circuit apart from being a device for demonstrating, studying andmodeling chaotic real world system, has been proposed to generatehyperchaotic circuit [T. Kapitaniak, L. O. Chua and G. Zhong,‘Experimental Hyperchaos in Coupled Chua's Circuits’, IEEE Trans. CAS-I,Vol 41, No.7, July 1994]. Referring to FIG. 3, it has been shown in ‘T.Kapitaniak, L. O. Chua and G. Zhong, ‘Experimental Hyperchaos in CoupledChua's Circuits’, IEEE Trans. CAS-I, Vol 41, No.7, July 1994’ that if aChua's (100) is coupled to a similar Chua's circuit (101) such that thenon-ground terminal of inductor is connected to the input of a voltagebuffer (10) whose output is connected to one of the terminal ofcontrolling resistor (11) and another end of the controlling resistor isconnected the non grounded terminal of another Chua's circuit (101), thesystem coupling is achieved by interconnecting ‘n’ Chua's circuit in thefashion described above and the last Chua's circuit of the ring iseither connected to the first Chua's circuit (100) or is left open. Thistype of coupling results in the following cases:

For the value of controlling resistor greater than specific value(called threshold value), all the Chua's circuit will synchronize witheach other. This value of controlling resistor depends on the number ofChua's circuit used in the chain. During this state the curve betweenthe state variable of one Chua's circuit to the corresponding statevariable in another Chua's circuit of the chain will be a straight line.

For the value of controlling resistor lesser than specific value (calledthreshold value), the Chua's circuit will loose synchronization witheach other and the system will undergo a state of hyper-chaos, which ismore sensitive to the initial condition. This value of controllingresistor depends on the number of Chua's circuit used in the chain.During this state the curve between the state variable of one Chua'scircuit to the corresponding state variable in another Chua's circuit ofthe chain will not be a straight line.

With such a proposal, a monolithic implementation of the hyperchaoticcircuit can be achieved by using any of the above-mentioned variants ofChua's circuit (i.e. prior arts of Chua's circuit can be used togenerate the hyperchaotic circuit using the above described scheme).However, use of any of these circuits will not only add the disadvantageof those proposal of Chua's circuit but also one voltage buffer and onefloating resistor per coupling is required which in turn make the finalcircuit bulky and inefficient in terms of power consumption.

OBJECT AND SUMMARY OF THE INVENTION

The object of the invention is to obviate above and other drawbacksassociated with the prior arts.

It is an object of the invention to design Chua's circuit using fourDual Output Current Conveyor (DO-CCII), two grounded resistors, threegrounded capacitors and three floating resistors.

Another object of the invention to use minimum grounded elements used sofar.

Yet another object of the invention is to have voltage across inductorin the form of current for current mode processing of the system withoutextra hardware requirement.

Further object of the invention to derive a Chua's diode using twogrounded resistors, two floating resistors and two MOCCIIs.

Yet another object of the invention to derive a Chua's diode using twogrounded resistors, two floating resistors and two DOCCIIs.

Further object of the invention to achieve current through Z-terminal ofDOCCII and MOCCII used in designing non-linear resistor.

Yet another object of the invention to design Chua's circuit usingMultiple Output Current Conveyor.

Further object of the invention to achieve grounded controlling resistorfor hyper-chaotic system designed by coupling of Chua's circuit which isdone by using the above derived current of voltage across inductor.

According to one embodiment an improved Chua's circuit is provided whileaccording to another embodiment, the CCII presented in [Seguin F. andFabre A.; ‘New second generation current conveyor with reduced parasiticresistance and band-pass filter application’, IEEE Trans. CAS-I, 2001,48,(6), pp. 781- 785] is modified to achieve a DOCCII and MOCCII asshown in FIG. 4. The schematic diagram of MOCCII and DOCCII used tofurther design Chua's chaotic circuit and it's coupling is as shown inFIG. 4B.

According to yet another embodiment, the above said DOCCII based Chua'scircuit can be implemented using MOCCII based Chua's circuit as shown inFIG. 5.

To achieve above objectives present invention provides an improvedChua's circuit providing current mode operation, access to all statevariables, minimum use of grounded passive elements, and freedom frompassive component matching comprising;

-   -   a multiple output current conveyer based inductor having one        grounded terminal,    -   a capacitor connected across the second terminal of said        inductor,    -   a resistor having one terminal connected to the second terminal        of said inductor,    -   the second terminal of said resistor connected to one terminal        of a second capacitor the other end of which is grounded, and    -   a pair of multiple output current conveyers connected together        to form a 2-terminal negative resistance having one terminal        connected to ground and the second terminal connected to the        second terminal of said resistance.

The said multiple output current conveyers are the multiple output(second generation) current conveyer (MO-CC II).

The said inductor comprises two multiple output current conveyers havingtheir x terminals grounded through resistors, a capacitor having oneterminal grounded and its second terminal connected to the z_(P1)terminal of the second multiple output current conveyer and in parallelto the y terminal of the first multiple output current conveyer whilethe y terminal of the second multiple output current conveyers and thez_(N1) terminal of the first multiple output current conveyers of arejoined together and which acts as one of the terminal of simulatedgrounded inductance.

The said 2-terminal negative resistance circuit comprises two multipleoutput current conveyers having their x terminals grounded throughresistors and their y terminals connected together and to the z_(P1)terminals of both multiple output current conveyers through twodifferent resistors.

An improved Chua's circuit providing current mode operation, access toall state variables, minimum use of grounded passive elements, andfreedom from passive component matching comprising;

-   -   a dual output current conveyer based inductor having one        grounded terminal,    -   a capacitor connected across the second terminal of said        inductor,    -   a resistor having one terminal connected to the second terminal        of said inductor, the second terminal of said resistor connected        to one terminal of a second capacitor the other end of which is        grounded, and    -   a pair of dual output current conveyers connected together to        form a 2-terminal negative resistance having one terminal        connected to ground and the second terminal connected to the        second terminal of said resistance.

The said dual output current conveyer is a dual output (secondgeneration) current conveyer.

The said inductor comprises two dual output current conveyers havingtheir x terminals grounded through resistors, a capacitor having oneterminal grounded and its second terminal connected to the z₊ terminalof the second dual output current conveyer and in parallel to the yterminal of the first dual output current conveyer while the y terminalof the second dual output current conveyers and the z terminal of thefirst dual output current conveyers of are joined together and whichacts as one of the terminal of simulated grounded inductance.

The said 2-terminal negative resistance circuit comprises two dualoutput current conveyers having their x terminals grounded throughresistors and their y terminals connected together and to the z₊terminals of both dual output current conveyers through two differentresistors.

An improved Chua's circuit for use in a hyperchaotic circuit comprisinga plurality of said Chua's circuit symmetrically coupled together toenable monolithic implementation of hyperchaotic circuit.

A method for improving a Chua's circuit to provide current modeoperation, access to all state variables, minimum use of groundedpassive elements, and freedom from passive component matching comprisingthe steps of:

-   -   simulating a grounded inductor using a multiple output current        conveyer,    -   connecting a capacitor connected across the second terminal of        said inductor,    -   connecting one terminal of a resistor to the second terminal of        said inductor,    -   connecting the second terminal of said resistor to one terminal        of a second capacitor the other end of which is grounded, and    -   providing a 2-terminal negative resistance device using a pair        of multiple output current conveyers.

BRIEF DESCRIPTION OF THE INVENTION WITH ACCOMPANYING DRAWINGS

The invention will now be described with reference to the accompanyingdrawings.

FIG. 1 shows the generalized Chua's circuit diagram

FIG. 2 shows the response of non-linear resistor (Chua's diode)

FIG. 3 shows the coupling of Chua's circuit

FIG. 4 shows the implementation of MOCCII

FIG. 5 shows the block diagram schematic of DOCCII and MOCCII

FIG. 6A shows the DOCCII based inductor

FIG. 6B shows the MOCCII based inductor

FIG. 7A shows the DOCCII based non-linear negative resistor

FIG. 7B shows the MOCCII based non-linear negative resistor

FIG. 8 shows the characteristic of thus designed non-linear negativeresistor

FIG. 9A shows the implementation of Chua's diode using DOCCII.

FIG. 9B shows the implementation of Chua's diode using MOCCII.

FIG. 10 shows the Chua's circuit using DOCCII

FIG. 11 shows the Chua's circuit using MOCCII and it's block schematic

FIG. 12 shows Double Scroll attractor derived from MOCCII based Chua'sCircuit

FIG. 13 shows the block schematic of MOCCII based hyperchaotic circuit

FIG. 14 shows the simulation results of hyperchaotic circuit validatingthe theory.

DETAILED DESCRIPTION OF THE INVENTION

The present invention provides a Chua's circuit using Dual Output(second-generation) Current Conveyer (DO-CC II) as one embodiment. Italso provides Chua's circuit implementation using Multiple OutputCurrent Conveyor and further it provides the design of a hyperchaoticcircuit using Multiple Output Current Conveyor based Chua's circuit asother embodiments.

FIG. 1 has already been discussed under the section “background of theinvention”.

FIG. 2 has also been discussed under the section “background of theinvention”

FIG. 3 shows the coupling of ‘n’ Chua's circuit to achievesynchronization. The system thus formed is used to solve the followingset of equations${C_{1}\frac{\mathbb{d}v_{C_{1}}^{(1)}}{\mathbb{d}t}} = {{G( {v_{C_{2}}^{(1)} - v_{C_{1}}^{(1)}} )} - {f( v_{C_{1}}^{(1)} )}}$${C_{2}\frac{\mathbb{d}v_{C_{2}}^{(1)}}{\mathbb{d}t}} = {{G( {v_{C_{1}}^{(1)} - v_{C_{2}}^{(1)}} )} + i_{L}^{(1)} + {f( v_{C_{1}}^{(1)} )} + {\frac{1}{R_{K}}( {v_{C_{2}}^{(2)} - v_{C_{2}}^{(1)}} )}}$${L\frac{\mathbb{d}v_{L}^{(1)}}{\mathbb{d}t}} = {- v_{C_{2}}^{(1)}}$${C_{1}\frac{\mathbb{d}v_{C_{1}}^{(2)}}{\mathbb{d}t}} = {{G( {v_{C_{2}}^{(2)} - v_{C_{1}}^{(2)}} )} - {f( v_{C_{1}}^{(2)} )}}$${C_{2}\frac{\mathbb{d}v_{C_{2}}^{(2)}}{\mathbb{d}t}} = {{G( {v_{C_{1}}^{(2)} - v_{C_{2}}^{(2)}} )} + i_{L}^{(2)} + {f( v_{C_{1}}^{(2)} )} + {\frac{1}{R_{K}}( {v_{C_{2}}^{(3)} - v_{C_{2}}^{(2)}} )}}$${L\frac{\mathbb{d}v_{L}^{(2)}}{\mathbb{d}t}} = {- v_{C_{2}}^{(2)}}$ ⋮${C_{1}\frac{\mathbb{d}v_{C_{1}}^{(n)}}{\mathbb{d}t}} = {{G( {v_{C_{2}}^{(n)} - v_{C_{1}}^{(n)}} )} - {f( v_{C_{1}}^{(n)} )}}$${C_{2}\frac{\mathbb{d}v_{C_{2}}^{(n)}}{\mathbb{d}t}} = {{G( {v_{C_{1}}^{(n)} - v_{C_{2}}^{(n)}} )} + i_{L}^{(n)} + {f( v_{C_{1}}^{(n)} )} + {\frac{1}{R_{K}}( {v_{C_{2}}^{({n + 1})} - v_{C_{2}}^{(n)}} )}}$${L\frac{\mathbb{d}v_{L}^{(n)}}{\mathbb{d}t}} = {- v_{C_{2}}^{(n)}}$

The dynamics of the system is dependent on the controlling resistorR_(K). This aspect has been discussed in prior art.

FIG. 4 shows the implementation of Multiple Output Current Conveyorwhich has been designed to by modifying the current conveyor proposed in[Seguin F. and Fabre A.; ‘New second generation current conveyor withreduced parasitic resistance and band-pass filter application’, IEEETrans. CAS-I, 2001,48,(6), pp. 781-785]. Herein one additional output z−is taken by using current mirrors. This scheme then corresponds toimplementation of DOCCII. Further, herein two additional outputs, onesimilar in behavior to Z+ and one similar in behavior to z− have beenadded. The final scheme as shown in figure realizes MOCCII.

FIG. 5 shows a schematic block diagram of DO-CC II and MOCCII. The DO-CCII has six terminals. Terminals VSS and VDD are the supply voltageterminals while the terminals x, y, z+ and z− allow different voltagesand current to flow in and out of the circuit. A basic DO-CC II has acurrent mirror circuit that reflects the current of terminal x in theterminal z+. The magnitude of the current through in z− terminal andthe, current through the z+ terminal is the same except for theirdirections. Also the voltages at the terminals x and y are the same andthe current through the terminal y is zero. This can be represented infollowing equations.v_(X)=v_(Y)  (1)i_(y)=0  (2)i _(Z+) =i _(X)  (3)i _(Z−) =−i _(X)  (4)

Where the subscript with current I and voltage V represents currents andvoltages in the respective terminals of the DO-CC II.

Further, the MO-CC II has eight terminals. Terminals VSS and VDD are thesupply voltage terminals while the terminals x, y, zp1, zp2 and zn1 andzn2 allow different voltages and currents to flow in and out of thecircuit.

A basic MO-CC II has a current mirror circuit that reflects the currentof terminal x in the terminal zp1. The magnitude of the current throughin zn1 terminal and the current through the zp1 terminal is the sameexcept for their directions. Also the voltages at the terminals x and yare the same and the current through the terminal y is zero. Further zp1and zp2 show similar behavior in terms of current movement. Similarlyzn1 and zn2 show similar behavior. This can be represented in followingequations.v_(X)=v_(Y)  (1)i_(y)=0  (2)i_(zp1)=i_(X)  (3)i _(zp2) =−i _(X)  (4)i_(zn1)=i_(X)  (5)i _(zn2) =−i _(X)  (6)

Where the subscript with current I and voltage V represents currents andvoltages in the respective terminals of the MO-CC II.

These characteristics of the DO-CC II and MO-CC II can be used torealize inductive properties as follows.

FIG. 6A shows DO-CC II based inductor. This inductor has two DO-CC II D₃and D₄ having their x terminals grounded through the resistors R_(L1),R_(L2). A capacitor C_(L) having its one terminal grounded and secondterminal coupled to z₊ terminal of DO-CC II D₃ and y terminal of DO-CCII D₄. The terminal z of D₄ and y of D₃ are joined together to connectthe one terminal of the input supply.

When a voltage v is applied at node 1, a similar voltage is induced atnode 2 that is at the x terminal of D₃ according to the equation 1 ofDO-CC II. Hence resulting in current i₂ through the resistor R_(L2)given byi₂=v/R_(L2).

Because of the relations 3 and 4, current through terminal x isreflected at terminals L and z₊ accordingly. Thus producing a voltageacross the capacitor C_(L) which is seen by the y terminal of D4 and isgiven by $v_{3} = {\int{\frac{v}{C_{L}R_{L2}}{\mathbb{d}t}}}$

The voltage V₃ is then induced at the x terminal of D₄ according to therelation 1 thus causing a current i5 through resistor R_(L1) that isthen, reflected in the terminals z+ and z− of D4. The current i5 throughthe resistor R_(L1) can be given as$i_{5} = {\frac{v_{3}}{R_{L1}} = {\int{\frac{v}{C_{L}R_{L2}R_{L1}}{\mathbb{d}t}}}}$

From FIG. 2 it is clear that the current i₅=i_(L) because of therelation 2, thus$i_{L} = {\int{\frac{v}{C_{L}R_{L2}R_{L1}}{\mathbb{d}t}}}$

From above following equation can be written:${C_{L}R_{L1}R_{L2}\frac{\mathbb{d}i_{L}}{\mathbb{d}t}} = v$

The characteristic equation of an inductor is given by:${L\frac{\mathbb{d}i_{L}}{\mathbb{d}t}} = v$

On comparing above two equations we get:L=C_(L)R_(L1)R_(L2)

Thus it can be said that the circuit shown is FIG. 6A is equivalent toan inductor of value determined by L=C_(L)R_(L1)R_(L2). Further it isimportant to note that the circuit provides terminals z+ and z− of D₄and D₃ respectively to tap current flowing through the inductor and thevoltage across capacitor (of tank circuit) in the form of current. To bespecific, whereas D₃ provides voltage across capacitor in the form ofcurrent, current through the inductor is available at z₊ terminal of D₄.

Similar results can be achieved by using MOCCII by replacing terminal xof DOCCII by terminal x of MOCCII, y of DOCCII by terminal y of MOCCII,z+ of DOCCII by terminal z1+ of MOCCII, z− of DOCCII by terminal z1− ofMOCCII, VDD of DOCCII by terminal VDD of MOCCII, VSS of DOCCII byterminal VSS of MOCCII and rest of the terminals of MOCCII i.e. z2+ andz2− remaining floating. Thus MOCC II based inductor is also realized asshown in FIG. 6B.

FIG. 7A shows a schematic block diagram of DOCC II based negativeresistor. The DOCC II has its x terminal connected to a resistance R3whose other terminal is connected to ground and has a resistance R1connected between its z+ and y terminals.

A Dual Output Current Conveyer based non-linear device basically worksin three different regions as depicted in IV Characteristics shown inFIG. 8. The region of operation can be classified as follows:

Negative saturation region is the region when the input voltage to thisnon-linear device is highly negative with respect to ground while thepositive saturation region, when the input voltage to this non-lineardevice is highly positive with respect to ground. The input voltage isthe voltage applied at the y terminal of the non-linear device describedin FIG. 7A. The Linear region is when the input voltage to thisnon-linear device is comparable to supply voltage.

In the negative or positive saturation regions when the input voltagesare either highly negative or positive, the z₊ terminal of the DOCCIIassumes a constant voltage (negative or positive respectively) thusallowing a current to flow through the resistance R3 exhibiting positiveresistive properties beyond a Break point (Bp) shown in FIG. 8.

In the linear region when the input voltage is comparable to the supplyvoltage, the current flowing inside z₊ terminal equals current through xterminal according to the relation 3. Also the voltage at the x terminaland the voltage at the y terminals are similar as per the relation 1 fora DOCCII. Thus current through the input supply terminal i_(in) is equalto the current through resistor R3 and hence the current ix through theterminal x. Thus the voltage at the terminal y can be given as:v _(y) =−i _(x) *R3Ori _(x) /v _(y)=−1/R3=m ₁

Where m₁ is the slope in the linear region.

Owing to its linearity in the three regions, positive linearity inpositive and negative saturation region and negative linearity in linearregion, the non-linear device behaves as a non-linear resistor withpositive resistance at the positive and negative saturation region andnegative resistance in the linear region.

Hence the DOCC II of FIG. 7A exhibits a negative resistance in thelinear region as shown in corresponding I-V Characteristics in the FIG.8.

Similar results can be achieved by using MOCCII by replacing terminal xof DOCCII by terminal x of MOCCII, y of DOCCII by terminal y of MOCCII,z+ of DOCCII by terminal zp1 of MOCCII, z− of DOCCII by terminal zn1 ofMOCCII, VDD of DOCCII by terminal VDD of MOCCII, VSS of DOCCII byterminal VSS of MOCCII and rest of the terminals of MOCCII i.e. zp2 andzn2 remaining floating. Thus MOCC II based negative resistor is alsorealized as shown in FIG. 7B.

FIG. 9A shows two DOCC II connected in parallel to achieve a variableslope in the linear region and specified break point as required for theChua's circuit. The total conductance of parallel combination of the twoDOCC II is a linear addition of the individual conductance of each DOCCII. Thus I-V characteristics of the parallel combination exhibit avariable slope as shown in FIG. 2. Note that this type of non-linearresistor is also called as Chua's diode.

Similar results can be achieved by using MOCCII by replacing terminal xof DOCCII by terminal x of MOCCII, y of DOCCII by terminal y of MOCCII,z+ of DOCCII by terminal zp1 of MOCCII, z− of DOCCII by terminal zn1 ofMOCCII, VDD of DOCCII by terminal VDD of MOCCII, VSS of DOCCII byterminal VSS of MOCCII and rest of the terminals of MOCCII i.e. zp2 andzn2 remaining floating. Thus MOCC II based Chua's diode is also realizedas shown in FIG. 9B.

Without limiting the scope of the invention to the discussed embodimentand the values thereof, the invention will now be discussed withreference to circuit shown in FIG. 10. A person skilled in art willappreciate that the invention can also be practiced with otherembodiments without deviating from the concept described hereinafter.

FIG. 10 shows a schematic block diagram of a Chua's circuit according toone embodiment of the present invention. The circuit shown is basicallya current mode implementation of Chua's circuit using a Dual Outputsecond generation Current Conveyor. The Chua's chaotic circuit accordingto the present invention comprises four Dual Output Second GenerationCurrent Conveyor D1, D2, D3 and D4. The DO-CC II D3 and D4 forming aninductor as described in FIG. 3 and D1, D2 forming a non-linearcomponent as described in the FIG. 9A. The capacitors C₁ and C₂ are thefirst and second energy storing elements of the Chua's circuit and areconnected to y terminals of D₃, D₂ respectively. Resistor R is thepassive component of the Chua's circuit and is connected to the yterminals of the D₃ and D₂. The terminals y of the D₁ and D₂ are coupledtogether and the terminals x of the D₁ and D₂ are connected to theground through resistive load R₄ and R₃. The terminals z+ of D₁ and D₂are connected to their y terminals through resistances R₁ and R₂ and they terminals of the D₂ and D₁ are connected as shown in FIG. 9A.

For this Chua's circuit the equation can be written as:${C_{2}\frac{\mathbb{d}V_{2}}{\mathbb{d}t}} = {{\frac{1}{R}( {V_{1} - V_{2}} )} - {g( V_{1} )}}$${C_{1}\frac{\mathbb{d}V_{1}}{\mathbb{d}t}} = {{\frac{1}{R}( {V_{2} - V_{1}} )} + i_{L}}$${C_{L}R_{L1}R_{L2}\frac{\mathbb{d}i_{L}}{\mathbb{d}t}} = {- V_{1}}$where,${g( V_{1} )} = {{m_{0}V_{1}} + {\frac{1}{2}{( {m_{0} - m_{1}} )\lbrack {{{V_{1} + B_{p}}} - {{V_{1} - B_{p}}}} \rbrack}}}$

The value of m₀ and m₁ are determined by resistor values R1, R2, R3, R4and the supply VSS1, VSS2, VDD1, VDD2.

FIG. 12 shows the results obtained by simulations of the above Chua'scircuit for values selected as follows. This is called the doublescroll-operating region.

Non Linear Resistance:

R₁=190 Ω, R₂=25.6 kΩ, R₃=2 kΩ, R₄=2.2 kΩ and VSS₁=−7V, VDD₁=7.8VVSS₂=−7.8V VDD₂=7V

Inductor:

C_(L)=100 nF, R_(L1)=400 Ω, R_(L2)=400 Ω.

Other Components:

C₁=100 nF, C₂=10 nF R=1.550k.

Similar results can be achieved by using MOCCII by replacing terminal xof DOCCII by terminal x of MOCCII, y of DOCCII by terminal y of MOCCII,z+ of DOCCII by terminal z1+ of MOCCII, z− of DOCCII by terminal z1− ofMOCCII, VDD of DOCCII by terminal VDD of MOCCII, VSS of DOCCII byterminal VSS of MOCCII and rest of the terminals of MOCCII i.e. z2+ andz2− remaining floating. Thus MOCC II based Chua's Circuit is alsorealized as shown in FIG. 11.

As described earlier the Chua's circuit can be coupled by using onevoltage buffer and one floating resistor using the scheme presented inFIG. 2. Herein it is actually solving the following equations of thesystem${C_{1}\frac{\mathbb{d}v_{C_{1}}^{(1)}}{\mathbb{d}t}} = {{G( {v_{C_{2}}^{(1)} - v_{C_{1}}^{(1)}} )} - {f( v_{C_{1}}^{(1)} )}}$${C_{2}\frac{\mathbb{d}v_{C_{2}}^{(1)}}{\mathbb{d}t}} = {{G( {v_{C_{1}}^{(1)} - v_{C_{2}}^{(1)}} )} + i_{L}^{(1)} + {f( v_{C_{1}}^{(1)} )} + {\frac{1}{R_{K}}( {v_{C_{2}}^{(2)} - v_{C_{2}}^{(1)}} )}}$${L\frac{\mathbb{d}v_{L}^{(1)}}{\mathbb{d}t}} = {- v_{C_{2}}^{(1)}}$${C_{1}\frac{\mathbb{d}v_{C_{1}}^{(2)}}{\mathbb{d}t}} = {{G( {v_{C_{2}}^{(2)} - v_{C_{1}}^{(2)}} )} - {f( v_{C_{1}}^{(2)} )}}$${C_{2}\frac{\mathbb{d}v_{C_{2}}^{(2)}}{\mathbb{d}t}} = {{G( {v_{C_{1}}^{(2)} - v_{C_{2}}^{(2)}} )} + i_{L}^{(2)} + {f( v_{C_{1}}^{(2)} )} + {\frac{1}{R_{K}}( {v_{C_{2}}^{(3)} - v_{C_{2}}^{(2)}} )}}$${L\frac{\mathbb{d}v_{L}^{(2)}}{\mathbb{d}t}} = {- v_{C_{2}}^{(2)}}$ ⋮${C_{1}\frac{\mathbb{d}v_{C_{1}}^{(n)}}{\mathbb{d}t}} = {{G( {v_{C_{2}}^{(n)} - v_{C_{1}}^{(n)}} )} - {f( v_{C_{1}}^{(n)} )}}$${C_{2}\frac{\mathbb{d}v_{C_{2}}^{(n)}}{\mathbb{d}t}} = {{G( {v_{C_{1}}^{(n)} - v_{C_{2}}^{(n)}} )} + i_{L}^{(n)} + {f( v_{C_{1}}^{(n)} )} + {\frac{1}{R_{K}}( {v_{C_{2}}^{({n + 1})} - v_{C_{2}}^{(n)}} )}}$${L\frac{\mathbb{d}v_{L}^{(n)}}{\mathbb{d}t}} = {- v_{C_{2}}^{(n)}}$

Here n is taken to be equal to 5. RK is the controlling resistor whosevalue describes the behavior of the entire system.

Present invention tries to achieve the solution of above equationswithout any additional hardware by proposing the scheme as shown in FIG.13.

Referring to FIG. 13, 11 of Chua's circuit (100) is connected to 17 ofsame Chua's circuit 100, 11 of Chua's circuit is also connected to 18 ofnext Chua's circuit (101), similarly 18 of Chua's circuit (100) isconnected with 11 of previous Chua's circuit (99), thereby forming aring using several Chua's circuit connected in similar and symmetricfashion. RL2 of each Chua's circuit acts as a controlling resistor andthe coupling is achieved without additional resistor or voltage buffer.The additional advantage of the present proposal of hyper chaoticcircuit is that the controlling resistor is grounded and hence can bebeneficial in easy monolithic implementation.

For the above stated values of components and RL1=300 Ohm and RL2=600ohm for Chua's circuit designed using MOCCII and using similar Chua'scircuit for coupling with the scheme as shown in FIG. 13, the system ofFIG. 13 thus derived is a hyperchaotic. This is proved by the fact thatnone of the Chua's circuit is in synchronization with each other as isreflected by the simulation results shown in FIG. 14.

Advantages

A DOCCII/MOCCII based implementation of Chua's circuit is presented. Thecircuit has advantages of grounded resistor and capacitor, minimumactive and passive component and accessibility of current acrossinductor. Moreover, apart from these advantages the voltage acrosscapacitor (of tank circuit) is also available in the form of current athigh impedance node. One of the applications of this current is ingenerating hyper-chaos in coupled Chua's circuit with reduced hardware.

The Chua's circuit uses a DOCC II/MOCC II based inductor and non-linearcomponent that allows tapping of current through the inductor andvoltage across the capacitor of tank circuit without requiring anyadditional hardware. The available third state variable can be observedand therefore it is possible to make more complex chaotic circuits usingthis additional information. Also the inductor of the present inventiondoes not use any additional components like capacitors and resistors ascompared to the prior arts.

Since the present invention does not use operational amplifiers ratherit uses current conveyors it operates in the current mode. Also theinvention does not require a precise component matching unlike priorarts that used op-amps for realizing Chua's circuit.

The invention also relates the use of available current in designingreduced hardware hyper-chaos circuit. The final hyper-chaotic circuitthus designed offers several advantages like minimum active and passivecomponents for coupled Chua's circuit hyper chaotic circuit, coupling ofChua's circuit without additional hardware as voltage buffer andfloating resistor, controlling resistor being grounded etc. Theseadvantages are non-existent in case the same coupling is achieved usingany of the prior art of Chua's circuit.

1. An improved Chua's circuit providing current mode operation, accessto all state variables, minimum use of grounded passive elements, andfreedom from passive component matching comprising; a multiple outputcurrent conveyer based inductor having one grounded terminal, acapacitor connected across the second terminal of said inductor, aresistor having one terminal connected to the second terminal of saidinductor, the second terminal of said resistor connected to one terminalof a second capacitor the other end of which is grounded, and a pair ofmultiple output current conveyers connected together to form a2-terminal negative resistance having one terminal connected to groundand the second terminal connected to the second terminal of saidresistance.
 2. An improved Chua's circuit as claimed in claim 1 whereinsaid multiple output current conveyers are the multiple output (secondgeneration) current conveyer (MO-CC II).
 3. An improved Chua's circuitas claimed in claim 1 wherein said inductor comprises two multipleoutput current conveyers having their x terminals grounded throughresistors, a capacitor having one terminal grounded and its secondterminal connected to the z_(P1) terminal of the second multiple outputcurrent conveyer and in parallel to the y terminal of the first multipleoutput current conveyer while the y terminal of the second multipleoutput current conveyers and the Z_(N1) terminal of the first multipleoutput current conveyers of are joined together and which acts as one ofthe terminal of simulated grounded inductance.
 4. An improved Chua'scircuit as claimed in claim 1 wherein said 2-terminal negativeresistance circuit comprises two multiple output current conveyershaving their x terminals grounded through resistors and their yterminals connected together and to the z_(P1) terminals of bothmultiple output current conveyers through two different resistors.
 5. Animproved Chua's circuit providing current mode operation, access to allstate variables, minimum use of grounded passive elements, and freedomfrom passive component matching comprising; a dual output currentconveyer based inductor having one grounded terminal, a capacitorconnected across the second terminal of said inductor, a resistor havingone terminal connected to the second terminal of said inductor, thesecond terminal of said resistor connected to one terminal of a secondcapacitor the other end of which is grounded, and a pair of dual outputcurrent conveyers connected together to form a 2-terminal negativeresistance having one terminal connected to ground and the secondterminal connected to the second terminal of said resistance.
 6. Animproved Chua's circuit as claimed in claim 6 wherein said dual outputcurrent conveyer is a dual output (second generation) current conveyer.7. An improved Chua's circuit as claimed in claim 6 wherein saidinductor comprises two dual output current conveyers having their xterminals grounded through resistors, a capacitor having one terminalgrounded and its second terminal connected to the z+ terminal of thesecond dual output current conveyer and in parallel to the y terminal ofthe first dual output current conveyer while the y terminal of thesecond dual output current conveyers and the z terminal of the firstdual output current conveyers of are joined together and which acts asone of the terminal of simulated grounded inductance.
 8. An improvedChua's circuit as claimed in claim 1 wherein said 2-terminal negativeresistance circuit comprises two dual output current conveyers havingtheir x terminals grounded through resistors and their y terminalsconnected together and to the z₊ terminals of both dual output currentconveyers through two different resistors.
 9. An improved Chua's circuitfor use in a hyperchaotic circuit as claimed in claims 1 comprising aplurality of said Chua's circuit symmetrically coupled together toenable monolithic implementation of hyperchaotic circuit.
 10. A methodfor improving a Chua's circuit to provide current mode operation, accessto all state variables, minimum use of grounded passive elements, andfreedom from passive component matching comprising the steps of:simulating a grounded inductor using a multiple output current conveyer,connecting a capacitor connected across the second terminal of saidinductor, connecting one terminal of a resistor to the second terminalof said inductor, connecting the second terminal of said resistor to oneterminal of a second capacitor the other end of which is grounded, andproviding a 2-terminal negative resistance device using a pair ofmultiple output current conveyers.